What is the definition of "free fall"?

1.
"An object is in "free fall" when the only force acting upon it is gravity".

2.
Is gravity in this case singular or plural?
Is the acting gravity the resultant force of all bodies in the universe?

3.
In theory, my own bodys gravitational force is acting on the object and thus it's not acually a free fall.
My teacher just says "no, read the definition of "free fall" in the textbook".
But it's in swedish and gravity is written like "the gravity", like it's a single force.

You're chasing your tail here. The force due to gravity is inversely proportional to the distance between the body falling and any other bodies nearby.

F = Gm1m2 / r2

Now, as an exercise, you can calculate the force of gravity between a falling mass of say, 1 kg, and you, and the earth, and the moon, and the sun, and whatever else you can think of. If you rank all of these various forces by magnitude, I think you'll see that the earth exerts the major influence on this falling body, and the forces exerted by all of these other bodies are insignificant.

Look at it this way: gravity is singular. If the only force acting is gravity, that's free fall. Now the cause of that gravity force can be one single body or a whole lot of them, but that doesn't matter.

Example: some point between earth and moon. Closer to the moon: free fall towards the moon. Closer to earth: free fall towards the earth. And there is a point where you're not accelerating wrt either of them. All three can be considered free fall (in the last case it'll just take a very, very long time before the fall ends .

Likewise, in SK's case, all the little forces may be insignificant, but that is not the point: they all originate from gravity, so it's free fall.